You estimate uncertainty from repeated measurements by calculating the mean and the standard deviation of your data, and then reporting the mean with the standard deviation as the uncertainty, or using the standard error if you want the uncertainty of the mean.

When you take the same measurement many times you get slightly different numbers, so first add them all up and divide by how many you took to get the average, which is the best single value to describe the measurements. Next, find how far each measurement is from that average, square those differences so they are all positive, add them up, and divide by the number of measurements minus one to get the variance; the square root of the variance is the standard deviation, which tells you how spread out the measurements are. If you want to know how precisely you have determined the average value, divide the standard deviation by the square root of the number of measurements to get the standard error, which usually becomes smaller the more times you measure. Always keep the units and use the same number of decimal places that match the uncertainty so your reported result looks correct and honest.

A common mistake is reporting too many digits and implying greater precision than the uncertainty supports, so round the average and uncertainty to matching sensible decimal places and report them together; another mistake is confusing standard deviation with standard error, so remember standard deviation describes spread of individual measurements while standard error estimates uncertainty of the mean; also people sometimes forget to use the sample formula that divides by the number of measurements minus one for small sets of data, so use that unbiased estimator when you have a sample rather than the whole population.

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Estimating Uncertainty from Repeated Measurements

1. Calculate the average of measurements
2. Find each measurement’s deviation from mean
3. Square deviations and sum them
4. Divide by measurements minus one
5. Take square root for standard deviation
6. Divide by sqrt(number of measurements)

Common Mistakes Estimating Measurement Uncertainty

1. Reporting too many digits
2. Confusing standard deviation and standard error
3. Using population formula for small samples

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